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  "body": "\n## Summary\n\nThe method of characteristics solves a first-order [[partial differential equation|GC5BDZ6QBC]] by reducing it to a system of [[ordinary differential equations|0NT64JV96A]].\nAlong special curves, the characteristics, the equation says how the unknown changes, so one integrates along these curves from the initial data.\nIt gives explicit solutions of transport, Hamilton–Jacobi and conservation law equations, and it explains why their solutions may break down in finite time.\nSee the [[definition|VDWKE1KK4H]].\n\n## Explanations\n\n### L1 human\n\nA first-order equation such as $u_t + c\\, u_x = 0$ says that $u$ does not change when one moves with speed $c$.\nSo one follows the lines $x = x_0 + ct$, and along each line $u$ keeps its initial value $u(x_0, 0)$.\nThis gives the solution $u(x, t) = u(x - ct, 0)$ at once.\nIn general the curves are not straight and $u$ may change along them, but one still only solves ordinary differential equations.\nWhen two characteristic curves carrying different values cross, the solution cannot stay smooth, and a shock forms.\n",
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